Methods and systems for concurrent paging of mobile users in cellular networks

ABSTRACT

Methods for concurrent paging, in a cellular system having a number of users, where the users are paged to determine a location of each user, are disclosed. The paging from one cell to one of the users occurs within one of a number of time slots. The time slot in which one of the users is paged from one of the cells is a characteristic of that cell and that user. An array encompassing the totality of the characteristics time slots constitutes a search schedule. One method comprises minimizing a characteristic function of paging performance, the function depending on the search schedule and the probability of finding a user in a cell, and, determining, from the minimization of the characteristic function, the search schedule. Two other methods for determining a search schedule, which are more computationally efficient and which reduce the paging cost, the simple heuristic and the conditional probability heuristic, are disclosed.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority of U.S. Provisional Application No.60/344,286, “Concurrent Search of Mobile Users in Cellular Networks”,filed on Dec. 27, 2001, which is incorporated by reference herein.

BACKGROUND OF THE INVENTION

This invention relates generally to wireless cellular networks, and,more particularly, to methods for finding the current cell of a wirelesscellular network in which a user resides.

In a cellular network, when a call to the mobile user arrives, amobility management scheme is responsible for finding the current cellin which a mobile user resides. Typically, a mobility management schemeconstitutes of a location update scheme and a paging scheme.

Over the last several years, the worldwide cellular communicationsmarket has experienced explosive growth. This growth has been driven bythe decrease in prices of cellular service and phones and improvedservice. In general, as the number of subscribers increases for a fixedradio spectrum allocation, the size of wireless coverage cells mustdecrease, in order to accommodate the higher subscriber densities. Ascells decrease in size, the problem of efficient location managementbecomes more difficult due to the additional signaling created by moresubscribers and more cells. The additional signaling useselectromagnetic spectrum bandwidth, a scarce resource.

In the last decade, many location update schemes were proposed.Basically, these schemes were either movement-based, timer-based,distance-based, profile-based, state-based, or velocity-based. Schemesthat use a hybrid of the above strategies were also proposed. It hasbeen proven that distance-based schemes achieve better performancecompared to movement-based schemes and timer-based schemes. Someproposed schemes suggested that a mobile user should register itslocation only when it enters some predefined cells, referred to asreporting centers. An information theoretic approach for location updateand derivation of probabilistic information about locations of mobileusers has been proposed (Amiya Bhattacharya and Sajal K. Das,“LeZi-Update: An Information-Theoretic Approach to Track Mobile Users inPCS Networks”, MOBICOM 1999. Pages 1–12).

A location area is composed of a number of cells. Some researchersassumed that a mobile user sends a location update message to the systemwhenever it enters a new location area and concentrated on the design ofan optimal location area. Kim and Lee proposed an integer-programmingmodel to find the optimal location area, which may take on an irregularshape. Other researchers assumed that location areas are given andfocused on the decision problem of whether a mobile user should send alocation update message when it enters a new location area.

Recently, a convex optimization problem was formulated to minimize thecosts of location update and paging in the movement-based locationupdate scheme (Jie Li, Hisao Kameda, and Keqin Li, “Optimal dynamicmobility management for PCS networks”, IEEE/ACM Transactions onNetworking, vol. 8, no. 3 p. 319–27, June 2000.). A continuousformulation of the problem of one-dimensional location area design hasbeen proposed to overcome the computational difficulty associated withthe original combinatorial formulation. An improved probabilisticlocation update scheme has been proposed. Probabilistic paging has alsobeen used for contention-free mobility management (Wing Ho A. Yuen andWing Shing Wong, “A contention-free mobility management scheme based onprobabilistic paging”, IEEE Transactions on Vehicular Technology, vol.50, no. 1, p. 48–58, January 2001).

There is an intrinsic tradeoff between location update and paging. Asthe frequency of location update increases, the location uncertaintydecreases and therefore the paging cost decreases. And on the contrary,when the frequency of location update decreases, both the locationuncertainty and paging cost increase. It is possible to see paging as amore fundamental operation than location update. However, as it has beenpointed out, “the majority of the research on location management hasactually focused on update schemes, assuming some obvious version of thepaging algorithm.”

The concept of dividing a location area into paging zones has beenpreviously described. Lyberopoulos (G. L. Lyberopoulos, J. G.Markoulidakis, D. V. Polymeros, D. F. Tsirkas and E. D. Sykas,“Intelligent paging strategies for third generation mobiletelecommunication systems”, IEEE Transactions on Vehicular Technology,vol. 44, no. 3, p. 543–553, August 1995) proposed to page the cell thata mobile user registered with most recently and then page all othercells in the location area if necessary. Rose and Yates (C. Rose and R.Yates, “Minimizing the average cost of paging and registration: Atimer-based method”, Wireless Networks, 2(2):109–116, June 1996) provedthat given the probabilistic information about the position of a mobileuser, to minimize the average paging cost, the cells with the higherprobabilities must be paged before the cells with the lowerprobabilities are paged. Krishnamachari, Gau, Wicker, and Haas (BhaskarKrishnamachari, Rung-Hung Gau, Stephen B. Wicker, Zygmunt J. Haas,“Optimal Sequential Paging in Cellular Networks”, IEEE VehicularTechnology Conference, Fall 2001) proposed an efficient algorithm tosolve the problem of minimizing the average paging cost under theworst-case paging delay constraint.

All the above works on paging focused on the problem of searching for asingle mobile user and assumed that some straightforward strategy ofsearching for multiple mobile users, such as sequential search, is used.

Rose et al.(C. Rose and R. Yates, “Minimizing the average cost of pagingand registration: A timer-based method”, Wireless Networks,2(2):109–116, June 1996) proposed a sequential paging scheme to locate asingle mobile user in the cellular network based on the probabilisticinformation of the location of the mobile user. The basic idea of such asystem resides in partitioning cells in the network into a number ofpaging zones and search for paging zones one by one. More precisely, thesystem pages the first paging zone in the first time slot. If the mobileuser is in the first paging zone, the mobile user is located and thesearch is aborted. Otherwise, the system pages the second paging zone,and so on.

In U.S. Pat. No. 6,181,945 (Jan. 30, 2001), Lee discloses a method forpaging a single user based on minimizing the paging cost.

All the above described work focuses on the problem of searching for asingle mobile user. However, it is more efficient for a multi-usersystem to service many paging requests simultaneously (or concurrently).A model has been proposed for the concurrent paging problem but it makesassumptions about the arrival times and/or the resident times of mobileusers (D. Goodman, P. Krishnan, B. Sugla, “Minimizing queuing delays andnumber of messages in mobile phone location”, Mobile Networks andApplications, Vol. 1, p. 39–48, 1996; C. Rose and R. Yates, “Ensemblepolling strategies for increased paging capacity in mobilecommunications networks”, ACM Wireless Networks, Vol. 3, No. 2, p.159–167, 1997, see also http://citeseer.nj.nec.com/rose96ensemble.html).

Consequently, a less restrictive and more efficient concurrent pagingscheme is needed. A concurrent paging scheme that minimizes the expectednumber of the required paging messages would be highly desirable.

It is therefore an object of this invention to provide a concurrentpaging scheme that minimizes the expected number of the required pagingmessages.

It is a further object of this invention to provide concurrent pagingschemes that can be easily computed and provide a reduction of theexpected number of the required paging messages.

SUMMARY OF THE INVENTION

The objects set forth above as well as further and other objects andadvantages of the present invention are achieved by the embodiments ofthe invention described hereinbelow.

Methods and systems for concurrent paging, in a cellular system having anumber of users, where the users are paged to determine a location ofeach user, are disclosed. The cellular system described is characterizedin the following manner. The paging to one of the users originates fromone of the cells. Any of the users can be found the cells with givenunique probabilities corresponding to the user and the cell. The pagingfrom one cell to one of the users occurs within one of a number of timeslots. The time slot in which one of the users is paged from one of thecells is specific of that cell and that user. An array encompassing thetotality of the specific time slots constitutes a search schedule. Inone of the time slots, only one of the users is paged from one of thecells.

One method of the present invention comprises minimizing acharacteristic function of paging performance, the function depending onthe search schedule and the probability of finding a user in a cell,and, determining, from the minimization of the characteristic function,the search schedule. The characteristic function can be, but is notlimited to, the average paging cost (the expected number of the requiredpaging messages) or the average paging delay.

Two other methods for determining a paging scheme (a search schedule),which are more computationally efficient and which reduce the pagingcost, the simple heuristic and the conditional probability heuristic,are disclosed. The simple heuristic method considers two users and twotime slots at a time, using two cumulative probabilities, calculatedutilizing the first half and second half of the cells respectively, toassign the time slot in which one of the two users is paged from one ofthe cells. The process is repeated until all pairs of users areconsidered. If the number of users is an odd number, the last user ispaged in the last time slot.

The conditional probability heuristic method produces a dynamic searchschedule. Starting with no users located and an updated probabilityarray containing the given unique probabilities, corresponding to a userand a cell, that the user is found in that cell, the method proceedsthrough all time slots updating the probability array as users are pagedor found. At every time slot, the user with the highest non-zeroprobability in a cell is paged from that cell. If the user is located,all probabilities for that user are set to zero in the subsequent timeslots. If a user is paged from a cell but not found, the probability offinding that user in that cell is set to zero. The above procedure isrepeated in the next time slot using the updated probability array untilall users are located (the updated probability array contains onlyzeroes) or last time slot is reached.

The present methods can be implemented as computer readable code and ina system utilizing the computer readable code.

For a better understanding of the present invention, together with otherand further objects thereof, reference is made to the accompanyingdrawings and detailed description and its scope will be pointed out inthe appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is an idealized graphical representation of a system comprisinga number of cells and a number of users;

FIG. 1B is a graphical and block representation of an embodiment of acellular system;

FIG. 2 is a flow chart of an embodiment of the simple heuristic methodof this invention for determining a paging scheme; and

FIG. 3 is a flow chart of an embodiment of the conditional probabilityheuristic method of this invention for determining a paging scheme;

FIG. 4 depicts the results of paging simulations for the reduction inpaging cost, over paging cost resulting from paging according to asequential search, when paging is performed according to a schedulegenerated by concurrent optimal search method, according to a schedulegenerated by the simple heuristic method, and according to a schedulegenerated by the conditional probability heuristic method, where thesequential search is conducted by paging every cell for t^(th) user inthe t^(th) time slot, for 4 users and 4 cells;

FIG. 5 depicts the results of the paging simulations for the reductionin paging cost, over paging cost resulting from paging according to asequential search generated as that of FIG. 4, when paging is performedaccording to a schedule generated by the simple heuristic method, andaccording to a schedule generated by the conditional probabilityheuristic method, for 8 users and 100 cells;

FIG. 6 depicts the results of paging simulations for the reduction inpaging cost, over paging cost resulting from paging according to asequential search generated as that of FIG. 4, when paging is performedaccording to a schedule generated by the simple heuristic method, andaccording to a schedule generated by the conditional probabilityheuristic method, for 16 users and 100 cells.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A cellular mobile communications system relies on a network of cells,areas of a radio transmission and reception, with a powerful radiotransmitter at the center, a base station. FIG. 1A is an idealizedgraphical representation of a system 10 comprising a number of cells 20and a number of user devices 30. Although cells 20 are represented bycircles in FIG. 1A, in a real situation, cells 20 have a shapereflecting the local radio propagation properties and such a shape canbe highly irregular. FIG. 1B is a graphical and block representation ofan embodiment of a cellular system including some of the components ofan embodiment of the system of FIG. 1A. Mobile user 30 is in the cellgenerated by a radio transmitter 40. A group of the radio transmitters40 is connected to a base station controller 50 that provides set up andcontrol for the radio channels. The base station controllers 50 areconnected to a mobile service switching center 60 that includes thefunctionality needed for location management. Each mobile switchingcenter 60 includes the computational capability and the databases neededfor location management and other services such as billing. While theembodiment depicted in FIG. 1B is characteristic of wireless telephonysystems, analogous embodiments exist for packet data systems.

Paging involves sending a message informing the user that an incomingmessage is pending. Referring to FIG. 1A, there are n cells 20 in thenetwork and k users 30 in the network. The probability that user i is incell j is given by p(i,j). The probability, p(i,j), can be obtained bymethods known in the art, for example, the methods given in U.S. Pat.No. 6,181,945, the information-theoretical approach used by Bhattacharya(Amiya Bhattacharya and Sajal K. Das, “LeZi-Update: AnInformation-Theoretic Approach to Track Mobile Users in PCS Networks”,MOBICOM 1999. Pages 1–12), or the methods proposed by Madhavapeddy (S.Madhavapeddy, K. Basu and A. Roberts, “Adaptive Paging Algorithms forCellular Systems”, Proc. Of the Fifth WINLAB Workshop on ThirdGeneration Wireless Networks. April 1995, p. 347–361). The paging in onecell to one of the users occurs within one of a number of time slots.

A simple (trivial) example serves to illustrate the difference in pagingcost between sequential paging (such as that disclosed in U.S. Pat. No.6,181,945) and concurrent paging, the object of this invention. In thisexample, the network includes k cells and n=k users, in which theprobability that user i is in cell j is1−ε when i=j,ε/(n−1) when i is not equal to j.

An efficient concurrent search scheme, which pages cell i to locatemobile user i in the first time slot, is able to locate all mobile usersat the end of the first time slot almost certainly. Therefore, theaverage number of required paging messages to locate n mobile users isapproximately n. In contrast, the straightforward sequential pagingscheme requires n² paging messages.

The index of the time slot in which user i is paged from cell j, x(i,j),is specific of that cell and that user. An array, X, encompassing thetotality of the specific time slots, in which the i,j element X _(i,j)is given by x(i,j), constitutes a search schedule. In any of the timeslots, only one of the users is paged in one of the cells. The s^(th)paging zone of user i, denoted by Z(i,s), is defined to be the set ofcell indices indicating those cells for which user i is paged in timeslot s. The probability that user i is inside its s-th paging zone,denoted by π(i,s) is given byΣp(i,j)where the summation is taken over all cells in Z(i,s). The number ofelements in the set Z(i,s) is denoted by m(i,s). Then, the averagepaging cost, C(P,X), defined as the expected number of required pagingmessages to locate mobile users, is given by

${C\left( {\underset{\_}{P},\underset{\_}{X}} \right)} = {\sum\limits_{i = 1}^{k}{\sum\limits_{s = 1}^{d}{{\pi\left( {i,s} \right)}{\sum\limits_{\alpha = 1}^{s}{m\left( {i,\alpha} \right)}}}}}$

An example can illustrate the above definitions. If the number of users(k) equals two (2), the number of cells (n) equals four (4), the numberof time slots (d) equals three (3),and the probabilities are given byp(1,1)=0.4, p(1,2)=0.3, p(1,3)=0.2, p(1,4)=0.1,p(2,1)=0.3, p(2,2)=p(2,3)=0.25, p(2,4)=0.2,then, the probability matrix is given by

$\underset{\_}{P} = \begin{bmatrix}{0.4\mspace{20mu} 0.3\mspace{31mu} 0.2\mspace{34mu} 0.1} \\{0.3\mspace{20mu} 0.25\mspace{20mu} 0.25\mspace{20mu} 0.2}\end{bmatrix}$

Assume that to locate mobile user one, the system pages cell 1 and cell2 in time slot 1, and pages cell 3 in time slot 2, if the user has notbeen located at the end of time slot 1. If the mobile user has not beenlocated at the end of time slot 2, the system pages cell 4 in time slot3. Then, Z(1,1)={1,2}, Z(1,2)={3}, and Z(1,3)={4}. Similarly, it isassumed that Z(2,1)={3,4}, Z(2,2)={1,2}, and Z(2,3)=empty set Then, thesearch schedule is

$\underset{\_}{X} = \begin{bmatrix}1 & 1 & 2 & 3 \\2 & 2 & 1 & 1\end{bmatrix}$

The values for probability that a user is inside a given paging zone aregiven by π(1,1)=0.4+0.3=0.7, π(1,2)=0.2 and π(1,3)=0.1. Similarly,π(2,1)=0.25+0.2=0.45, π(2,2)=0.3+0.25=0.55 and π(2,3)=0. The number ofelements in each of the sets Z(i,s) is given by m(1,1)=2, m(1,2)=1,m(1,3)=1, m(2,1)=m(2,2)=2 and m(2,3)=0. Then, the average paging cost,C(P,X), also referred to as the expected number of required pagingmessages to locate mobile users, is given byC(P,X)=(0.7*2+0.2*3+0.1*4)+(0.45*2+0.55*4+0*4*4) =5.5. If thestraightforward sequential paging scheme is used,

$\underset{\_}{X} = \begin{bmatrix}1 & 1 & 1 & 1 \\2 & 2 & 2 & 2\end{bmatrix}$then, C(P,X)=2*4=3.

If the number of time slots is greater than or equal to the product ofthe number of cells (n) and the number of users (d≧k·n), the concurrentsearch problem is identical to a collection of k independent sequentialpaging problems. In the embodiments disclosed below, the moreinteresting case of d<k·n is considered. Specifically, in theembodiments disclosed below, the number of time slots equals the numberof users (d=k).

The average paging cost depends on the search schedule and theprobability of finding a user in a cell through the dependence of x onthe probability and the dependence of m on the index of the time slotsthat constitute the search schedule. An optimal concurrent search (orpaging) schedule, which is an object of this invention, is obtained byminimizing the average paging cost. The optimization problem can besolved by the “brute force” search algorithm, which would simply searchfor the whole state space of valid search schedules. It should beapparent that other algorithms could be found that would be capable ofsolving the optimization problem or approximate a solution to theoptimization problem.

A similar optimal concurrent search (or paging) schedule can be found byminimizing the minimum average paging delay. The concurrent search (orpaging) schedule can be found by minimizing the minimum average pagingdelay, D(P, X), given by

${D\left( {\underset{\_}{P},\underset{\_}{X}} \right)} = {\left( {1/k} \right){\sum\limits_{i = 1}^{k}{\sum\limits_{s = 1}^{d}{{\pi\left( {i,s} \right)}*s}}}}$An optimal concurrent search (or paging) schedule is obtained byminimizing the average paging delay.

The optimal concurrent search obtained by minimizing the average pagingcost reduces the expected number of required paging messages to locatemobile users (the average paging cost) over a sequential search. Thenormalized reduction of the average paging cost for a k times nprobability matrix P, denoted by r*(k,n,P), is defined as follows:

${r^{*}\left( {k,n,P} \right)} = \frac{\left( {{C_{seq}(P)} - {C_{k}^{*}(P)}} \right)}{C_{seq}(P)}$where C_(seq)(P) is the average paging cost for a sequential search, andC*_(k)(P) is the optimal average paging cost for a concurrent search. Ithas shown by Gau and Haas (U.S. Provisional Application No. 60/344,286)that even when mobile users appear equally in all of the cells in thenetwork, the concurrent search scheme is able to reduce the averagepaging cost without increasing the average paging delay or theworst-case paging delay.

Simple Heuristic Algorithm

Although the “brute force” search algorithm guarantees an optimalsolution, its computational complexity is exponential and therefore itis not desirable. The simple heuristic algorithm, disclosed below, is alinear-time algorithm and it guarantees the normalized reduction of theaverage paging cost by at least 16.6%.

FIG. 2 is a block diagram of an embodiment of the simple heuristicmethod for determining a paging scheme. When the number of cells is n, amid-value of the number of cells (n_(mid)) is equal to half the totalnumber of cells (n/2) when the number of cells is even and is equal tohalf the total number of cells plus one( (n/2)+1) when the number ofcells is odd. Referring to FIG. 2, starting with i=1 (where i is theindex of the user), a consecutive pair of users and time slots isselected (step 100, FIG. 2). The pair comprises a first user and a firsttime slot and a second user and a second time slot. A first probabilitysum, Φ₁, equal to a sum of the probabilities corresponding to the firstuser and each of the cells ranging from one to the mid-value cell , iscomputed (step 110, FIG. 2),

$~{\Phi_{1} = {\sum\limits_{j = 1}^{n_{mid}}{p\left( {i_{first},j} \right)}}}$where i_(first) is initially one (1). Then, a second probability sum,Φ₂, equal to a sum of the probabilities corresponding to the second userand each of the cells ranging from the initial one to the mid-value cellis computed (step 110, FIG. 2),

$\Phi_{2} = {\sum\limits_{j = 1}^{n_{mid}}{p\left( {i_{second},j} \right)}}$where i_(second) is initially two (2). The first probability sum and thesecond probability sum are compared (step 120, FIG. 2). If the firstprobability sum is greater than or equal to the second probability sum,the paging (search) schedule is obtained (step 130, FIG. 2) byperforming the following steps:

-   -   1) for cells, ranging from the initial cell to a number equal to        the mid value, setting each characteristic number of the time        slot, corresponding to the first user and to each cell in the        range, equal to the identifying number of the first time slot        and setting each characteristic number of the time slot,        corresponding to the second user and to each cell in the range,        equal to the identifying number of the second time slot, and    -   2) for cells ranging from a number equal to the mid value plus        one to a number equal to the total number of cells, setting each        characteristic number of the time slot, corresponding to the        first user and to each cell in the range, equal to the        identifying number of the second time slot and setting each        characteristic number of the time slot, corresponding to the        second user and to each cell in the range, equal to the        identifying number of the first time slot.        If the first probability sum is less than the second probability        sum, the paging (search) schedule is obtained (step 140, FIG. 2)        by performing the following steps:    -   1) for cells ranging from the initial cell to the mid value,        setting each characteristic number of the time slot,        corresponding to the first user and to each cell in the range,        equal to the identifying number of the second time slot and        setting each characteristic number of the time slot,        corresponding to the second user and to each cell in the range,        equal to the identifying number of the first time slot; and    -   2) for cells with an identifying number ranging from a number        equal to the mid value plus one to a number equal to the total        number of cells, setting each characteristic number of the time        slot, corresponding to the first user and to each cell in the        range, equal to the identifying number of the first time slot        and setting each characteristic number of the time slot,        corresponding to the second user and to each cell in the range,        equal to the identifying number of the second time slot.

The above process is repeated (step 150, FIG. 2) for the nextconsecutive pair of users and time slots until an integer number ofpairs equal to half the number of users (truncated to an integer value).If the number of users is odd (step 160, FIG. 2), the last user (afterconsidering all the pairs) is paged simultaneously from all cells in thelast time slot. The paging schedule (step 170, FIG. 2) for the last useris obtained by setting each characteristic number of the paging timeslot, corresponding to the last user and to each cell, equal to theidentifying number of the last time slot.

An example to illustrate the simple heuristic algorithm is given below.For four users being paged in four cells, the probability matrix, givingthe probability of finding a user in a cell, is

$P = \begin{matrix}0.4 & 0.3 & 0.2 & 0.1 \\0.3 & 0.25 & 0.25 & 0.2 \\0.1 & 0.2 & 0.3 & 0.4 \\0.2 & 0.25 & 0.25 & 0.3\end{matrix}$

Considering the first two users, Φ₁ is given by 0.4+0.3=0.7, and Φ₂ isgiven by 0.3+0.25=0.55. Since Φ₁ is greater than Φ₂, x(1,1)=1, x(1,2)=1,x(2,1)=2, x(2,2)=2, x(1,3)=2, x(1,4)=2, x(2,3)=1, x(2,4)=1. Then,considering the next two users, Φ₁ is given by 0.1+0.2=0.3, and Φ₂ isgiven by 0.2+0.25=0.45. Thus, Φ₂ is greater than Φ₁, and x(3,1)=4,x(3,2)=4, x(3,3)=3, x(3,4)=3, x(4,1)=3, x(4,2)=3, x(4,3)=4, x(4,4)=4.Then, the paging (search) schedule is

$X = \begin{matrix}1 & 1 & 2 & 2 \\2 & 2 & 1 & 1 \\4 & 4 & 3 & 3 \\3 & 3 & 4 & 4\end{matrix}$

If the number of users were odd, five (5) for example, then, the lastrow in the paging schedule would have all fives (5). The normalizedreduction of the average paging cost for a k times n probability matrixP, is defined as above except that C*_(k)(P) is now the average pagingcost obtained from the results of the heuristic algorithm. It has shownby Gau and Haas (U.S. Provisional Application No. 60/344,286) that a16.6% to 25% reduction in average paging cost is obtained when theconcurrent search schedule is calculated by means of the simpleheuristic algorithm and that use of a concurrent search schedulecalculated by means of the simple heuristic algorithm does not increasethe paging delay.

Conditional Probability Heuristic Algorithm To further reduce theaverage paging cost, a conditional probability heuristic algorithm toobtain a dynamic schedule is disclosed herein below. Unlike a staticschedule, a dynamic schedule is not fully determined at the beginning ofthe first time slot. Instead, a dynamic schedule is obtained based onthe paging results of the previous time slots. FIG. 3 is a block diagramof an embodiment of the conditional probability heuristic method fordetermining a paging scheme. In every time slot, before paging, a uniquestate is defined for every user, s(i,t). The state is an indicator ofwhether the user has been located, and has a value of zero if the userhas not been located and a value of 1 otherwise. Also, in every timeslot, before paging, an updated probability array, A(t) is calculatedreflecting the certainty of finding the users that have been located.

At the initial time slot, the updated probability array is set equal tothe initial probability array, P. Also, at the initial time slot thestate for every user is set to indicate that no users have been located(data block 200, FIG. 3). A time slot counter, tn, is established andset to 1. For each cell, at the current time slot, select, according toa selection criterion, one user, φ(j,tn), to be paged (step 210, FIG.3). In one embodiment, the selection criterion used is the maximumprobability criterion. That is, at each cell, the user with maximumprobability, from the updated probability array, is selected to bepaged. At each cell j (j=1,n), the selected user, φ(j,tn), is paged(step 220, FIG. 3). From the results of paging, the state andprobability array are updated. If a user is found, the state of thatuser, s(i,tn+1) is set to one (1), otherwise the state remains as it wasprevious to paging (step 230, FIG. 3). The probability matrix is updated(step 240, FIG. 3) as follows:

-   -   a) if a user has been located, the state of the user is        s(i,tn+1)=1, then the probabilities for that user in all cells        are set to zero; that is A(tn+)_(i,j)=0 for j=1,n, where n is        the number of cells;    -   b) if the user has not been located, the probabilities for those        cells from which the user has been paged are set to zero; that        is,        A(tn+1)_(i,j)=0 for all j such that φ(j,tn)=i;    -   c) the non zero values of the probabilities have to be adjusted        to renormalize them or, in other words, adjust them for the        known information (also known as the conditional probability        calculation), that is

${A\left( {{tn} + 1} \right)}_{i,j} = {{\frac{{A({tn})}_{i,j}}{1 - {\sum\limits_{{\alpha:{\phi{({\alpha,{tn}})}}} = i}{A({tn})}_{i,\alpha}}}{for}\mspace{14mu}{all}\mspace{14mu} j\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu}{\phi\left( {j,{tn}} \right)}} \neq i}$

-   -    (for those cells from which that user was not paged).

If all probabilities are zero (step 250, FIG. 3), then all users havebeen located and the algorithm ends (steps 260, 280, FIG. 3). If thetime slot is the last time slot (equal to the number of users, asdetermined in step 280, FIG. 3), the algorithm ends. If allprobabilities are not zero and the time slot is not the last time slot,the algorithm, increments the time slot counter, moves to next time slot(step 270, FIG. 3) and repeats the above steps.

The Conditional Probability Heuristic Algorithm can be illustrated bythe following example with 3 users and 4 cells. The initial probabilitymatrix is

$P = \begin{matrix}0.4 & 0.3 & 0.2 & 0.1 \\0.3 & 0.25 & 0.25 & 0.2 \\0.1 & 0.2 & 0.3 & 0.4\end{matrix}$

At the first time slot, tn=1, the probability matrix is equal to theinitial probability matrix and the state of all users is zero (step 200,FIG. 3).

According to the maximum probability criterion, in time slot one (1),page user 1 in cell 1, page user 1 in cell 2, page user 3 in cells 3 and4 (step 210, FIG. 3). A search schedule row vector then isX(1, i)=[1 1 3 3]

Users are paged and User 1 is found in this time slot (Step 220, FIG.3). Then, upon updating the state, s(1, 2)=1 and all other states arezero (step 230, FIG. 3) Updating the probability matrix (step 240, FIG.3),

-   -   A(2)_(i,j)=0 for j=1,4, and    -   A(2)_(3,3)=A(2)_(3,4)=0 since user 3 was paged from cells 3 and        4,

$\begin{matrix}{{{A(2)}_{3,1} = {\frac{0.1}{1 - \left( {0.3 + 0.4} \right)} = {\frac{1}{3}\mspace{14mu}{and}}}},{similarly},{{{A(2)}_{3,2} = {\frac{0.2}{0.3} = \frac{2}{3}}};}} \\{{A(2)} = \begin{matrix}0 & 0 & 0 & 0 \\0.3 & 0.25 & 0.25 & 0.2 \\\frac{1}{3} & \frac{2}{3} & 0 & 0\end{matrix}}\end{matrix}$

The elements of A(2) are not all zero (step 250, FIG. 3). Thus, tnincreases to 2 (step 270, FIG. 3). Repeating steps 210 and 220,according to the maximum probability criterion, in time slot two (2),page user 3 in cell 1, page user 3 in cell 2, page user 2 in cells 3 and4. A search schedule row vector then isX(2, i)=[3 3 2 2]

User 3 is found in time slot 2. Then, upon updating the state, s(3, 3)=1and s(2,3) is zero (step 230, FIG. 3). Updating the probability matrix(step 240, FIG. 3),

-   -   A(3)_(3,j)=0 for j=1, 4 since user 3 was located;    -   A(3)_(2,3)=A(3)_(2,4)=0 since user 2 was paged from cells 3 and        4;

$\begin{matrix}{{{A(3)}_{2,1} = {\frac{0.3}{1 - \left( {0.25 + 0.2} \right)} = {\frac{0.3}{0.55} = {{\frac{6}{11}\mspace{14mu}{and}\mspace{14mu}{A(3)}_{2,2}} = {\frac{0.25}{0.55} = \frac{5}{11}}}}}};} \\{{A(3)} = \begin{matrix}0 & 0 & 0 & 0 \\\frac{6}{11} & \frac{5}{11} & 0 & 0 \\0 & 0 & 0 & 0\end{matrix}}\end{matrix}$

The elements of A(3) are not all zero (step 250, FIG. 3). Thus, tnincreases to 4 (step 270, FIG. 3). Repeating steps 210 and 220,according to the maximum probability criterion, in time slot three (3),page user 2 in cells 1 and 2. A search schedule row vector then isX(3, i)=[2 2]

User 2 is found in time slot 3. Updating the probability matrix (step240, FIG. 3),

-   -   A(4)_(2,1)=A(4)_(2,2)=0 since user 2 was located; and

${A(4)} = \begin{matrix}0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{matrix}$Since the elements of A(4) are all zero (step 250, FIG. 3), thealgorithm terminates.

The advantages of the methods of this invention can be best seen fromthe simulations presented below. In these simulations, for a network ofn=w² cells, the probability that mobile user t is in a cell indexed by(α,β) is given by

$\frac{r^{d{({\alpha,\beta,\alpha^{*},\beta^{*}})}}}{\sum\limits_{i = 0}^{w - 1}{\sum\limits_{j = 0}^{w - 1}r^{d{({{\mathbb{i}},j,\alpha^{*},\beta^{*}})}}}}$where r, the decay factor, is a real number between 0 and 1, α*,β* areuniformly distributed random integers variables between 0 and (w−1), andd is the distance between two cells, given byd(α₁, β₁, α₂, β_(2)=|α) ₁−α₂|+|β₁−β₂|.

When the decay factor is very close to 1 (0.99), the users appear withalmost equal probability in the cells. Under these conditions, thereduction in “paging cost” over a sequential search is the smallest.Alternatively, when the decay factor is closest to zero, the concurrentsearch produces the largest reduction in “paging cost” over a sequentialsearch. In order to conduct the simulations, the “quasi-random”variables are generated by means known in the art. FIGS. 4, 5, and 6depict the results of the simulations when the sequential search isconducted by paging every cell for t^(th) user in the t^(th) time slot.

As can be seen from FIG. 4, the optimal search provides the largestreduction in the paging cost and the difference in a reduction in pagingcost between the optimal search, the simple heuristic search and theconditional probability heuristic search is always less than 15%. As canbe seen from a FIG. 5, for 8 users and 100 cells, the conditionalprobability heuristic search provides a very large reduction in thepaging cost, but the simple heuristic search provides a 35% reduction inthe paging cost for small values of the decay factor (non uniformprobability distribution of users among the cells). Similar results areobtained for 16 users and 100 cells, as can be seen from FIG. 6.

As can be inferred by the simulations, the methods of this invention canbe implemented as computer readable code that it is typically embodiedin a computer usable (readable) medium. The computer readable codeembodied in a computer readable medium can cause one or many processorsin a system to generate a search schedule, which is then used to pageusers in a cellular network. In one embodiment, the one or manyprocessors and the one or many computer readable media are located inthe message switching center 60 of FIG. 1B. The search schedule, if theprobabilities are known beforehand, could be generated and then storedin a computer readable medium. A system with one or many processors andone or several schedules stored in one or many computer readable mediacan be used to page users in a cellular network.

The methods of this invention avoid the high average paging cost due tosearching users one by one. Although sequential paging schemes have beenproposed to reduce the average paging cost, such as the method disclosedin U.S. Pat. No. 6,181,945, the concurrent search schedule generated bythe methods of this invention presents an improvement over thesequential paging schedule. The sequential paging schedule is designedto locate a single user, while the concurrent search schedule of thisinvention is designed to locate a number of users simultaneously. Thesequential paging scheme reduces the average paging cost at the expenseof increasing the worst case paging delay. The concurrent searchschedule of this invention does not increase the worst-case pagingdelay.

Each computer program within the scope of the claims below may beimplemented in any programming language, such as assembly language,machine language, a high-level procedural programming language, or anobject-oriented programming language. The programming language may be acompiled or interpreted programming language.

Each computer program may be implemented in a computer program producttangibly embodied in a computer-readable storage device for execution bya computer processor. Method steps of the invention may be performed bya computer processor executing a program tangibly embodied on acomputer-readable medium to perform functions of the invention byoperating on input and generating output.

Common forms of computer-readable media include, for example, a floppydisk, a flexible disk, hard disk, magnetic tape, or any other magneticmedium, a CDROM, any other optical medium, punched cards, paper tape,any other physical medium with patterns of holes, a RAM, a PROM, andEPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrierwave, or any other medium from which a computer can read.

Elements and components described herein may be further divided intoadditional components or joined together to form fewer components forperforming the same functions.

Although the invention has been described with respect to variousembodiments, it should be realized this invention is also capable of awide variety of further and other embodiments within the spirit andscope of the appended claims.

1. A method for concurrently paging, in a cellular system having aplurality of users, the users being paged to determine a location ofeach user, the paging to one of the plurality of users originating fromone of a plurality of cells, any of the users being located in any ofthe cells with a given unique probability corresponding to the user andthe cell, the paging from the cell to one of the users occurring withinone of a plurality of time slots, the time slot in which one of theusers is paged in one of the cells being a characteristic of the onecell and the one user, and wherein, in one of said time slots, only oneof the users is paged from one of the cells, the method comprising thesteps of: minimizing a characteristic function of paging performance,said function depending on a search schedule and the probability offinding a user in a cell; wherein said search schedule comprises a twodimensional array of indices of a time slot in which one user is pagedfrom one cell; determining, from the minimization of the characteristicfunction, an optimal search schedule; whereby users from said pluralityof users are paged simultaneously.
 2. The method of claim 1 wherein thestep of minimizing said characteristic function comprises obtaining aminimum average paging cost.
 3. The method of claim 1 wherein the stepof minimizing said characteristic function comprises obtaining a minimumaverage paging delay.
 4. A method for concurrently paging in a cellularsystem having a plurality of users, the users being paged to determine alocation of each user, the paging to one of the plurality of usersoriginating from one of a plurality of cells, any of the users beinglocated in any of the cells with a given unique probabilitycorresponding to the user and the cell, the paging from the cell to oneof the users occurring within one of a plurality of time slots, the timeslot in which one of the users is paged in one of the cells being acharacteristic of the one cell and the one user, and, wherein, in one ofsaid time slots, only one of the users is paged from one of the cells,the method comprising the steps of: minimizing a characteristic functionof paging performance: wherein the step of minimizing a characteristicfunction of paging performance comprises the steps of: A) selecting apair of users, a first user and a second user, from the plurality ofusers; B) selecting a pair of time slots, a first time slot and a secondtime slot, from the plurality of time slots; C) determining, from theunique probability of any of the users being located in any of thecells, which of said first and said second users has a higherprobability of being located in a pre-selected range from the pluralityof cells; D) if it is more probable that said first user be located insaid pre-selected range, performing the following steps: paging saidfirst user from said first time slot in each cell from said pre-selectedrange; paging said second user from said second time slot in each cellfrom said pre-selected range; paging said first user from said secondtime slot in each cell from another pre-selected range of said pluralityof cells; and, paging said second user from said first time slot in eachcell from said another pre-selected range; E) if it is more probablethat said second user be located in any cell from said pre-selectedrange, performing the following steps: paging said first user from saidsecond time slot in each cell from said pre-selected range; paging saidsecond user from said first time slot in each cell from saidpre-selected range; paging said first user from said first time slot ineach cell from said another pre-selected range; and, paging said seconduser from said second time slot in each cell from said anotherpre-selected range; F) repeating steps A through E for a next pair ofusers and a next pair of time slots until all pairs of users areselected; whereby, from the minimization of the characteristic function,a search schedule is determined; wherein said search schedule comprisesa two dimensional array of indices of a time slot in which one user ispaged from one cell.
 5. The method of claim 4 further comprising thestep of: G) paging a last user from the plurality of users from a lasttime slot if a total number of users from the plurality of users is oddand a total number of time slots from the plurality of time slot is odd.6. A method for concurrently paging, in a cellular system having aplurality of users, the users being paged to determine a location ofeach user, the paging to one of the plurality of users originating fromone of a plurality of cells, any of the users being located in any ofthe cells with a given unique probability corresponding to the user andthe cell, the paging from the cell to one of the users occurring withinone of a plurality of time slots, the time slot in which one of theusers is paged in one of the cells being a characteristic of the onecell and the one user, and, wherein, in one of said time slots, only oneof the users is paged from one of the cells, the method comprising thesteps of: minimizing a characteristic function of paging performance:wherein the step of minimizing a characteristic function of pagingperformance comprises the steps of: A) obtaining a probability arrayhaving as initial value an array comprising a totality of the uniqueprobabilities; B) selecting an initial time slot from the plurality oftime slots; C) selecting, for each one cell from the plurality of cells,one user to be paged from the plurality of users; D) paging the selecteduser in said each one cell; E) determining from said paging whether theselected user has been located; F) updating the probability arrayaccording to said paging; G) proceeding to a next time slot from theplurality of time slots; H) repeating the steps C through G unless apredetermined condition is satisfied; and determining, from theminimization of the characteristic function, a search schedule; whereinsaid search schedule comprises a two dimensional array of indices of atime slot in which one user is paged from one cell.
 7. The method ofclaim 6 wherein said predetermined condition comprises said next timeslot being a last time slot from said plurality of time slots or allusers from the plurality of users being located.
 8. The method of claim6 wherein said selection criterion, according to which a user isselected to paged in each cell, comprises selecting the user withmaximum probability.
 9. A computer program product comprising: acomputer usable medium having computer readable code embodied thereinfor concurrently paging, in a cellular system having a plurality ofusers, the users being paged to determine a location of each user, thepaging to one of the plurality of users originating from one of aplurality of cells, any of the users being located in any of the cellswith a given unique probability corresponding to the user and the cell,the paging from the cell to one of the users occurring within one of aplurality of time slots, the time slot in which one of the users ispaged in one of the cells being a characteristic of the one cell and theone user, and, wherein, in one of said time slots, only one of the usersis paged from one of the cells, said computer program productcomprising: computer readable code means for causing a computer systemto minimize a characteristic function of paging performance, saidfunction depending on a search schedule and a probability of finding auser in a cell; wherein said search schedule comprises a two dimensionalarray of indices of a time slot in which one user is paged from onecell; and computer readable code means for causing a computer system todetermine, from the minimization of the characteristic function, anoptimal search schedule; wherein said optimal search schedule enablessimultaneous paging.
 10. The computer program product of claim 9 whereinsaid computer readable code means for causing a computer system tominimize a characteristic function comprise computer readable code meansfor causing a computer system to obtain a minimum average paging cost.11. The computer program product of claim 9 wherein said computerreadable code means for causing a computer system to minimize acharacteristic function comprise computer readable code means forcausing a computer system to obtain a minimum average paging delay. 12.A computer program product comprising: a computer usable medium havingcomputer readable code embodied therein for concurrently paging, in acellular system having a plurality of users, the users being paged todetermine a location of each user, the paging to one of the plurality ofusers originating from one of a plurality of cells, any of the usersbeing located in any of the cells with a given unique probabilitycorresponding to the user and the cell, the paging from the cell to oneof the users occurring within one of a plurality of time slots, the timeslot in which one of the users is paged in one of the cells being acharacteristic of the one cell and the one user, and, wherein, in one ofsaid time slots, only one of the users is paged from one of the cells,the computer program product: comprising: computer readable code meansfor causing a computer system to minimize a characteristic function ofpaging performance; wherein said computer readable code means forcausing said computer system to minimize a characteristic function ofpaging performance cause said computer system to: A) select a pair ofusers, a first user and a second user, from the plurality of users; B)select a pair of time slots, a first time slot and a second time slot,from the plurality of time slots; C) determine, from the uniqueprobability of any of the users being located in any of the cells, whichof said first and said second users has a higher probability of beinglocated in a pre-selected range from the plurality of cells; D) if it ismore probable that said first user be located in said pre-selectedrange, perform the following steps: page said first user from said firsttime slot in each cell from said pre-selected range; page said seconduser from said second time slot in each cell from said pre-selectedrange; page said first user from said second time slot in each cell fromanother pre-selected range of said plurality of cells; and, page saidsecond user from said first time slot in each cell from said anotherpre-selected range; E) if it is more probable that said second user belocated in any cell from said pre-selected range, perform the followingsteps: page said first user from said second time slot in each cell fromsaid pre-selected range; page said second user from said first time slotin each cell from said pre-selected range; page said first user fromsaid first time slot in each cell from said another pre-selected range;and, page said second user from said second time slot in each cell fromsaid another pre-selected range; F) repeat steps A through E for a nextpair of users and a next pair of time slots until all pairs of users areselected; computer readable code means for causing said computer systemto determine, from the minimization of the characteristic function, asearch schedule; wherein said search schedule comprises a twodimensional array of indicies of a time slot in which one user is pagedfrom one cell which enables concurrent paging.
 13. The computer programproduct of claim 12 wherein said computer readable code means forcausing said computer system to minimize a characteristic function ofpaging performance further cause said computer system to: G) page a lastuser from the plurality of users from a last time slot if a total numberof users from the plurality of users is odd and a total number of timeslots from the plurality of time slot is odd.
 14. A computer programproduct comprising: a computer usable medium having computer readablecode embodied therein for concurrently paging, in a cellular systemhaving a plurality of users, the users being paged to determine alocation of each user, the paging to one of the plurality of usersoriginating from one of a plurality of cells, any of the users beinglocated in any of the cells with a given unique probabilitycorresponding to the user and the cell, the paging from the cell to oneof the users occurring within one of a plurality of time slots, the timeslot in which one of the users is paged in one of the cells being acharacteristic of the one cell and the one user, and, wherein, in one ofsaid time slots, only one of the users is paged from one of the cells,the computer program product comprising: computer readable code meansfor causing a computer system to minimize a characteristic function ofpaging performance; wherein said computer readable code means forcausing said computer system to minimize a characteristic function ofpaging performance are capable of causing said computer system to: A)obtain a probability array having as initial value an array comprising atotality of the unique probabilities; B) select an initial time slotfrom the plurality of time slots; C) select for each one cell from theplurality of cells, one user to be paged from the plurality of users; D)page the selected user in said each one cell; E) determine from saidpaging whether the selected user has been located; F) update theprobability array according to said paging; G) proceed to a next timeslot from the plurality of time slots; H) repeat the steps C through Gunless a predetermined condition is satisfied; computer readable codemeans for causing said computer system to determine, from theminimization of the characteristic function, a search schedule; whereinsaid search schedule comprises a two dimensional array of indices of atime slot in which one user is paged from one cell, which enablesconcurrent paging.
 15. The computer program product of claim 14 whereinsaid selection criterion, according to which a user is selected to pagedin each cell, comprises selecting the user with maximum probability. 16.The computer program product of claim 14 wherein said predeterminedcondition comprises said next time slot being a last time slot from theplurality of time slots or all users from the plurality of users beinglocated.
 17. A system for concurrently paging, in a cellular systemhaving a plurality of users, the users being paged to determine alocation of each user, the paging to one of the plurality of usersoriginating from one of a plurality of cells, any of the users beinglocated in any of the cells with a given unique probabilitycorresponding to the user and the cell, the paging from the cell to oneof the users occurring within one of a plurality of time slots, the timeslot in which one of the users is paged in one of the cells being acharacteristic of the one cell and the one user, and, wherein, in one ofsaid time slots, only one of the users is paged from one of the cells,the system comprising: at least one processor; at least one computerreadable medium, having computer readable code embodied therein, saidcode causing the at least one processor to: minimize a characteristicfunction of paging performance, said function depending on a searchschedule and a probability of finding a user in a cell; wherein saidsearch schedule comprises a two dimensional array of indices of a timeslot in which one user is paged from one cell: determine, from theminimization of the characteristic function, the an optimal searchschedule; wherein said optimal search schedule enables simultaneouspaging.
 18. The system of claim 17 wherein said minimizing acharacteristic function comprises obtaining a minimum average pagingcost.
 19. The system of claim 17 wherein said minimizing acharacteristic function comprises obtaining a minimum average pagingdelay.
 20. A system for concurrently paging in a cellular system havinga plurality of users, the users being paged to determine a location ofeach user, the paging to one of the plurality of users originating fromone of a plurality of cells, any of the users being located in any ofthe cells with a given unique probability corresponding to the user andthe cell, the paging from the cell to one of the users occurring withinone of a plurality of time slots, the time slot in which one of theusers is paged in one of the cells being a characteristic of the onecell and the one user, and, wherein, in one of said time slots, only oneof the users is paged from one of the cells, the system comprising: atleast one processor: at least one computer readable medium, havingcomputer readable code embodied therein, said code causing the at leastone processor to: A) select a pair of users, a first user and a seconduser, from the plurality of users; B) selecting a pair of time slots, afirst time slot and a second time slot, from the plurality of timeslots; C) determining, from the unique probability of any of the usersbeing located in any of the cells, which of said first and said secondusers has a higher probability of being located in a pre-selected rangefrom the plurality of cells; D) if it is more probable that said firstuser be located in said pre-selected range, performing the followingsteps: paging said first user from said first time slot in each cellfrom said pre-selected range; paging said second user from said secondtime slot in each cell from said pre-selected range; paging said firstuser from said second time slot in each cell from another pre-selectedrange of said plurality of cells; and, paging said second user from saidfirst time slot in each cell from said another pre-selected range; E) ifit is more probable that said second user be located in any cell fromsaid pre-selected range, performing the following steps: paging saidfirst user from said second time slot in each cell from saidpre-selected range; paging said second user from said first time slot ineach cell from said pre-selected range; paging said first user from saidfirst time slot in each cell from said another pre-selected range; and,paging said second user from said second time slot in each cell fromsaid another pre-selected range; F) repeating steps A through E for anext pair of users and a next pair of time slots until all pairs ofusers are selected; wherein a search schedule is determined; whereinsaid search schedule comprises a two dimensional all of indices of atime slot in which one user is paged from one cell, which enablesconcurrent paging.
 21. The system of claim 20 wherein said code iscapable of further causing said at least one processor to: G) page alast user from said plurality of users from a last time slot if a totalnumber of users from said plurality of users is odd and a total numberof time slots from said plurality of time slot is odd.
 22. A system forconcurrently paging, in a cellular system having a plurality of users,the users being paged to determine a location of each user, the pagingto one of the plurality of users originating from one of a plurality ofcells, any of the users being located in any of the cells with a givenunique probability corresponding to the user and the cell, the pagingfrom the cell to one of the users occurring within one of a plurality oftime slots, the time slot in which one of the users is paged in one ofthe cells being a characteristic of the one cell and the one user, and,wherein, in one of said time slots, only one of the users is paged fromone of the cells, the system comprising: at least one processor; atleast one computer readable medium, having computer readable codeembodied therein, said code causing the at least one processor to: A)obtain a probability array having as initial value an array comprising atotality of the unique probabilities; B) select an initial time slotfrom the plurality of time slots; C) select for each one cell from theplurality of cells, one user to be paged from the plurality of users; D)page the selected user in said each one cell; E) determine from saidpaging whether the selected user has been located; F) update theprobability array according to said paging; G) proceed to a next timeslot from the plurality of time slots; H) repeat the steps C through Gunless a predetermined condition is satisfied; wherein a search scheduleis determined; wherein said search schedule comprises a two dimensionalarray of indices of a time slot in which one user is paged from onecell, which enables concurrent paging.
 23. The system of claim 22wherein said selection criterion, according to which a user is selectedto paged in each cell, comprises selecting the user with maximumprobability.
 24. The system of claim 22 wherein said predeterminedcondition comprises said next time slot being a last time slot from theplurality of time slots or all users from the plurality of users beinglocated.
 25. A concurrent paging schedule for concurrently paging, in acellular system having a plurality of users, the users being paged todetermine a location of each user, the paging to one of the plurality ofusers originating from one of a plurality of cells, any of the usersbeing located in any of the cells with a given unique probabilitycorresponding to the user and the cell, the paging from the cell to oneof the users occurring within one of a plurality of time slots, the timeslot in which one of the users is paged in one of the cells being acharacteristic of the one cell and the one user, and, wherein, in one ofsaid time slots, only one of the users is paged from one of the cells,the concurrent paging schedule obtained by the steps of: minimizing acharacteristic function of paging performance, said function dependingon a search schedule and a probability of finding a user in a cell;wherein said search schedule comprises a two dimensional array ofindices of a time slot in which one user is passed from one cell,determining, from the minimization of the characteristic function, anoptimal search schedule; wherein said optimal search schedule enablesconcurrent paging.
 26. The paging schedule of claim 25 wherein the stepof minimizing a characteristic function comprises obtaining a minimumaverage paging cost.
 27. The paging schedule of claim 25 wherein thestep of minimizing a characteristic function comprises obtaining aminimum average paging delay.
 28. A concurrent paging schedule forconcurrently paging, in a cellular system having a plurality of users,the users being paged to determine a location of each user, the pagingto one of the plurality of users originating from one of a plurality ofcells, any of the users being located in any of the cells with a givenunique probability corresponding to the user and the cell, the pagingfrom the cell to one of the users occurring within one of a plurality oftime slots, the time slot in which one of the users is paged in one ofthe cells being a characteristic of the one cell and the one user, and,wherein, in one of said time slots, only one of the users is paged fromone of the cells, the concurrent paging schedule obtained by the stepsof: A) selecting a pair of users, a first user and a second user, fromthe plurality of users; B) selecting a pair of time slots, a first timeslot and a second time slot, from the plurality of time slots; C)determining, from the unique probability of any of the users beinglocated in any of the cells, which of said first and said second usershas a higher probability of being located in a pre-selected range fromthe plurality of cells; D) if it is more probable that said first userbe located in said pre-selected range, performing the following steps:paging said first user from said first time slot in each cell from saidpre-selected range; paging said second user from said second time slotin each cell from said pre-selected range; paging said first user fromsaid second time slot in each cell from another pre-selected range ofsaid plurality of cells; and, paging said second user from said firsttime slot in each cell from said another pre-selected range; E) if it ismore probable that said second user be located in any cell from saidpre-selected range, performing the following steps: paging said firstuser from said second time slot in each cell from said pre-selectedrange; paging said second user from said first time slot in each cellfrom said pre-selected range; paging said first user from said firsttime slot in each cell from said another pre-selected range; and, pagingsaid second user from said second time slot in each cell from saidanother pre-selected range; F) repeating steps A through E for a nextpair of users and a next pair of time slots until all pairs of users areselected whereby said concurrent paging schedule enables concurrentpaging.
 29. The concurrent paging schedule as in claim 28 obtained bythe further steps of: G) paging a last user from the plurality of usersfrom a last time slot if a total number of users from the plurality ofusers is odd and a total number of time slots from the plurality of timeslot is odd.
 30. A concurrent paging schedule for concurrently paging,in a cellular system having a plurality of users, the users being pagedto determine a location of each user, the paging to one of the pluralityof users originating from one of a plurality of cells, any of the usersbeing located in any of the cells with a given unique probabilitycorresponding to the user and the cell, the paging from the cell to oneof the users occurring within one of a plurality of time slots, the timeslot in which one of the users is paged in one of the cells being acharacteristic of the one cell and the one user, and, wherein, in one ofsaid time slots, only one of the users is paged from one of the cells,the concurrent paging schedule obtained by the steps of: A) obtaining aprobability array having as initial value an array comprising a totalityof the unique probabilities; B) selecting an initial time slot from aplurality of time slots; C) selecting, for each one cell from theplurality of cells, one user to be paged from the plurality of users; D)paging the selected user in said each one cell; E) determining from saidpaging whether the selected user has been located; F) updating theprobability array according to said paging; G) proceeding to a next timeslot from the plurality of time slots; H) repeating the steps C throughG unless a predetermined condition is satisfied, wherein said concurrentpaging schedule comprises a two dimensional array of indices of a timeslot in which one of the users is paged in one of the cells, whichenables concurrent paging.
 31. The paging schedule of claim 30 wherein aselection criterion, according to which a user is selected to paged ineach cell, comprises selecting the user with maximum probability. 32.The paging schedule of claim 30 wherein said predetermined conditioncomprises said next time slot being a last time slot from the pluralityof time slots or all users from the plurality of users being located.